Substitution Method Calculator

The substitution method solves simultaneous equations by expressing one variable in terms of the other and inserting that expression into the second equation. For a two-variable linear system, both equations must be true at the same point. A unique solution is an ordered pair whose x and y values satisfy each equation. On a graph, that pair is the intersection of the two lines. Write the equations as a1x + b1y = c1 and a2x + b2y = c2. To substitute manually, choose a variable with a convenient nonzero coefficient. For example, solve the first equation for x, producing x = (c1 - b1y) / a1. Replace x in the second equation with that expression, solve the resulting one-variable equation for y, and then substitute y back into either original equation to obtain x. Finally, check both values in both equations. This calculator computes the mathematically equivalent determinant formulas so that it remains reliable regardless of which coefficient is easiest to isolate. The determinant is a1b2 - a2b1. When it is nonzero, the lines have different slopes and exactly one solution. The coordinates are x = (c1b2 - c2b1) divided by the determinant and y = (a1c2 - a2c1) divided by the determinant. These values are the same ones produced by correctly performed substitution. A zero determinant requires special interpretation. If one complete equation is a multiple of the other, the equations describe the same line and every point on that line is a solution. The system is dependent and has infinitely many solutions. If the variable coefficients are proportional but the constants are not, the lines are parallel and never meet. That inconsistent system has no solution. The result panel distinguishes these cases rather than attempting division by zero. Systems of linear equations model prices and quantities, mixtures, distances and rates, budgets, coordinate geometry, and many other relationships. Keep negative signs with their coefficients, and enter zero when a variable is absent from an equation. Decimal coefficients are accepted, though exact fractions may produce repeating decimal outputs. Rounding is limited to the display; the calculations use full JavaScript numeric precision. Always verify a high-stakes result by replacing x and y in the original equations.